Algebra

The derivative without infinitesimals and the mean value theorem

At the end of the eighteenth century, calculus worked very well and nobody could properly explain why. Leibniz’s $dx$ were “infinitely small” quantities: sometimes they were treated as if they were different from zero, so that one could divide by them, and right afterwards as if they were zero, so that one could discard them. Bishop Berkeley had mocked this in 1734, calling them “ghosts of departed quantities”. Lagrange was bothered by it. In 1797 he published a book whose subtitle is a whole programme:

October 14, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Galois at 15, drawing by an unknown artist: source page · public domain. Galois drawn by his brother Alfred, published in 1848: source page · public domain. The letter to Auguste Chevalier, 29 May 1832: source page · public domain. The Louis-le-Grand college, Paris, photo by Celette: source page · CC BY-SA 4.0. Blog figures The figure of the six symmetries of the triangle is an original illustration. The table in the post was checked by a program that composes the permutations of the corners. The figure has no text and serves every language of the site.

October 16, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Abel, drawing by Johan Gørbitz, Paris, 1826: source page · public domain. First volume of Crelle’s Journal, Berlin, 1826: source page · public domain. Abel’s grave at Froland, photo by Karl Ragnar Gjertsen: source page · CC BY 4.0. Abel monument in Oslo, sculpture by Gustav Vigeland, photo by Wolfmann: source page · CC BY-SA 4.0. Blog figures The figure of the lemniscate divided into eight equal arcs is an original illustration. The points were computed with the formula $r=\sqrt{\sqrt2-1}$ derived in the post. The figure has no text and serves every language of the site.

October 16, 2026 Read

Lagrange multipliers: optimising under a constraint

Almost every real decision is a maximum or minimum problem with a constraint. A factory wants the greatest profit but has a limited amount of raw material. A can must hold 350 ml using as little aluminium as possible. A pendulum moves however it likes, as long as the bob stays attached to the string. Lagrange found a single method for all these problems. It appears in the Analytical Mechanics of 1788, to deal with the ties of mechanical systems, and in the Theory of Analytic Functions of 1797, as a general rule of calculation.

October 14, 2026 Read

The Icosian game and Hamiltonian cycles

In 1856, still busy with non-commutative algebras, Hamilton invented the icosian calculus: three symbols $\iota$, $\kappa$ and $\lambda$ obeying the rules $$\iota^2 = 1, \qquad \kappa^3 = 1, \qquad \lambda^5 = 1, \qquad \lambda = \iota\kappa.$$Today we would say they generate the group of rotations of the icosahedron, the solid with twenty triangular faces. Hamilton presented the calculus at a meeting of the British Association in Dublin in 1857, and noticed that it described paths along the edges of the dodecahedron, the solid with twelve pentagonal faces and twenty vertices.

October 27, 2026 Read

Lagrange's legacy

Lagrange is the lofty pyramid of the mathematical sciences. — Napoleon Bonaparte, according to his contemporaries Lagrange left no anecdotes of precocious genius like Pascal, nor a body of work the size of Euler’s. He left something else: a way of doing mathematics. Where others drew, he calculated; where others found a trick for each problem, he looked for the rule that explained all the tricks. Almost everything he did is still in the textbooks under his name: the Euler–Lagrange equation, the Lagrangian, the Lagrange points, Lagrange’s theorem, the Lagrange remainder, Lagrange multipliers.

October 14, 2026 Read

Image credits for Lagrange

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Joseph-Louis Lagrange, engraving by an unknown artist: source page · public domain. Medallion with Lagrange’s profile, bronze by Pierre-Jean David d’Angers, Musées d’Angers: source page · CC0. Title page of the Méchanique analitique, Paris, 1788: source page · public domain. Statue of Lagrange in Turin, photo by Twice25, 2009: source page · CC BY-SA 3.0. Blog figures The cover, the diagram of the five Lagrange points and the graph of curves of equal area with the constraint line are original illustrations, with versions in each of the site’s languages. The diagram of the points is not to scale: L1 and L2 are in reality a hundred times closer to the Earth than the Sun is.

October 14, 2026 Read

Errors that make sense: false rules

Ask a class what $(a+b)^2$ is and some will answer $a^2 + b^2$. Ask what $\sqrt{9 + 16}$ is and someone will say $3 + 4 = 7$. These errors are so common, and so similar from one school to another, that they cannot be chance. They follow a rule. The rule that stretches too far In 1982 the American researcher Marilyn Matz studied the algebra errors of high-school students and proposed that many of them come from false rules (malrules) built out of true ones. The most productive of all is the distributive law:

October 25, 2026 Read

References and credits

Further reading Usiskin, Zalman. Conceptions of school algebra and uses of variables. In: Coxford, A. F.; Shulte, A. P. (eds.). The Ideas of Algebra, K–12. Reston: NCTM, 1988, pp. 8–19. Küchemann, Dietmar. Algebra. In: Hart, K. M. (ed.). Children’s Understanding of Mathematics: 11–16. London: John Murray, 1981, pp. 102–119. Clement, John; Lochhead, Jack; Monk, George S. Translation difficulties in learning mathematics. The American Mathematical Monthly, 88(4), 1981, pp. 286–290. MacGregor, Mollie; Stacey, Kaye. Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33, 1997, pp. 1–19. Philipp, Randolph A. The many uses of algebraic variables. The Mathematics Teacher, 85(7), 1992, pp. 557–561. Kieran, Carolyn. The learning and teaching of school algebra. In: Grouws, D. A. (ed.). Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan, 1992, pp. 390–419. Kleiner, Israel. Evolution of the function concept: a brief survey. The College Mathematics Journal, 20(4), 1989, pp. 282–300. Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. On the cossic signs, the letters of Viète and Descartes, $f(x)$ and Herschel’s $\sin^{-1}$. Heath, Thomas L. Diophantus of Alexandria: A Study in the History of Greek Algebra. Cambridge: Cambridge University Press, 1910. Viète, François. In artem analyticem isagoge. Tours: Jamet Mettayer, 1591. l’Hôpital, Guillaume de. Analyse des infiniment petits, pour l’intelligence des lignes courbes. Paris: Imprimerie Royale, 1696. The older texts on functions on this site supplied the idea of the function as a sentence with a hole in it, used in chapter 1 and taken up again here.

October 15, 2026 Read

How to read a letter: a guide and a glossary

This chapter showed that a letter can do at least five different jobs and that the writing hardly ever says which. The guide below gathers the questions that help to find out. Five questions for a letter Does it hide a value or stand for any value? If it hides one, it is an unknown, and the task is to find it. If it stands for any, the sentence is a general law or an identity. Does it move during the problem? If so, it is a variable, and it is worth asking what it depends on and what depends on it. Does it stay still while the others move? Then it is a parameter or a constant. It is worth asking: fixed for ever, like $\pi$, or only in this problem, like the $a$ in $ax^2$? Is it tied to an operator? If it appears in a $\Sigma$, an $\int$, a $\forall$ or a $\mapsto$, it is a bound letter, and it can be swapped for another without changing anything. Which word in the text announces the role? “Find”, “for every”, “let”, “when”, “given”. If none appears, the conventions of the subject decide, and it is good to make them explicit. A full reading of y = ax² + bx + c $y$ and $x$: variables. $x$ is the independent one, $y$ the dependent one. Together they run through the points of a parabola. $a$, $b$ and $c$: parameters. Fixed during the problem, they choose which parabola. The $a$ must be non-zero, or the curve becomes a straight line. Change the question and the roles change. In “where does the parabola cross the $x$-axis?”, we set $y=0$, and $x$ becomes an unknown. In “which parabola passes through $(1,2)$, $(2,3)$ and $(3,6)$?”, the parameters become the unknowns, and $x$ and $y$ become data: $a+b+c=2$, $4a+2b+c=3$, $9a+3b+c=6$, which give $a=1$, $b=-2$, $c=3$. No letter is a dummy. They all matter: swapping $a$ for $k$ requires swapping it throughout the text. One short line, three different roles, and each question rearranges everything.

October 15, 2026 Read

When the letter misleads: students, professors and apples

In 1981 three researchers at the University of Massachusetts, John Clement, Jack Lochhead and George Monk, set first-year engineering students a simple problem: Write an equation using the variables $S$ and $P$ to represent the following statement: “There are six times as many students as professors at this university.” Use $S$ for the number of students and $P$ for the number of professors. About 37% got it wrong. And almost all of those who got it wrong wrote the same thing:

October 15, 2026 Read

Dummy letters: the k of the sum and the t of the integral

These two expressions have the same value: $$\sum_{k=1}^{4}k^2=1+4+9+16=30,\qquad\sum_{j=1}^{4}j^2=1+4+9+16=30.$$Swapping $k$ for $j$ changed nothing. The letter is there only to mark where the calculation passes; once the sum is done, it disappears. Letters like these are called dummy, or bound. They are the opposite of unknowns: an unknown keeps a value that matters; a dummy letter keeps no value at all once the operation is over. Free and bound Now compare:

October 15, 2026 Read